Extend polynomial identities to complex numbers for higher-degree polynomial work.
Build the unique monic real polynomial from roots
Problem
Build and expand the monic real-coefficient polynomial with roots \(2~+~i\) and \(2~-~i\).
Big Picture
What this problem is really about
Turn each listed root r into the factor x minus r. To keep coefficients real, pair every nonreal root with its conjugate before expanding; their product becomes a real quadratic centered at the common real part. Multiplying all factors with leading coefficient one produces the unique monic real polynomial, which can be checked by substituting the original roots.
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